Introduction:

In two's complement representation, the most significant bit (MSB) is the sign bit, where 0 represents a positive number and 1 represents a negative number. To find the product of two numbers in two's complement, we can use the traditional binary multiplication method, followed by the proper interpretation of the result.

Given Information:

X = 11111011
Y = 00001010

Step 1: Converting to decimal:

We first need to convert the two's complement numbers into decimal form.

X = -5 (as the MSB is 1, representing a negative number)
Y = 10

Step 2: Multiplication:

Performing binary multiplication on X and Y:

```
11111011 (-5)
x 00001010 (10)
___________
11111011
00000000
11111011
+ 11111011
___________
110011110 (-78)
```

Step 3: Converting back to two's complement:

The result of the binary multiplication is 110011110. To interpret this as a two's complement number, we need to check the MSB.

Since the MSB of the result is 1, it represents a negative number. To obtain the two's complement representation, we need to perform the following steps:

1. Invert all the bits except the MSB:
00110001

2. Add 1 to the inverted value:
00110001 + 1 = 00110010

3. Assign the negative sign to the result:
-00110010

Therefore, the product of X and Y in two's complement representation is -78, which can be represented as '11001110' in binary.

Conclusion:

The value of the product of X and Y in two's complement is '11001110'. This result is obtained by performing binary multiplication on the two numbers and converting the result back to two's complement representation.